Accelerated Algorithms for Constrained Nonconvex-Nonconcave Min-Max Optimization and Comonotone Inclusion

TL;DR

This work advances constrained comonotone min-max optimization and comonotone inclusion by developing two first-order algorithms, composite-EAG and composite-FEG, that achieve the optimal $O\left(\frac{1}{T}\right)$ last-iterate convergence under broad comonotonicity conditions. The authors extend the Extra Anchored Gradient (EAG) and Fast Extra Gradient (FEG) frameworks to handle constraints and negative comonotonicity, proving both rate guarantees and point convergence via potential-function analyses and merging-path arguments with Optimized Halpern’s Method. A key technical contribution is the use of carefully designed resolvent-based updates that incorporate a proximal term $c_k\in A(z_k)$ to realize $(F+A)$ as a single operator, along with tailored potential functions that certify $O\left(\frac{1}{T}\right)$ rates. The results generalize prior unconstrained results, widen the admissible $\rho$-range for comonotone inclusion, and provide a unifying analytical approach that may aid the study of other accelerated algorithms in nonconvex-nonconcave min-max settings.

Abstract

We study constrained comonotone min-max optimization, a structured class of nonconvex-nonconcave min-max optimization problems, and their generalization to comonotone inclusion. In our first contribution, we extend the Extra Anchored Gradient (EAG) algorithm, originally proposed by Yoon and Ryu (2021) for unconstrained min-max optimization, to constrained comonotone min-max optimization and comonotone inclusion, achieving an optimal convergence rate of $O\left(\frac{1}{T}\right)$ among all first-order methods. Additionally, we prove that the algorithm's iterations converge to a point in the solution set. In our second contribution, we extend the Fast Extra Gradient (FEG) algorithm, as developed by Lee and Kim (2021), to constrained comonotone min-max optimization and comonotone inclusion, achieving the same $O\left(\frac{1}{T}\right)$ convergence rate. This rate is applicable to the broadest set of comonotone inclusion problems yet studied in the literature. Our analyses are based on simple potential function arguments, which might be useful for analyzing other accelerated algorithms.

Theorems & Definitions (59)

• Proposition 2.1
• Example 1: Min-Max Optimization
• Remark 3.1
• Theorem 3.2: Last-iterate convergence rate of \ref{['composite-EAG']}
• Lemma 3.3: Upper bound of $V_1$
• Theorem 3.4: Approximately non-increasing potential
• Lemma 3.5
• Corollary 3.6
• Theorem 3.7: Theorem 30.1 of bauschke2017convex, Theorem 2.1 of lieder2021convergence
• Theorem 3.8: Merging path between \ref{['composite-EAG']} and \ref{['OHM']}